\subsubsection{Force torque wrench space}

Rather than using only the forces to build a 3D wrench space, the torques can be included as well, resulting in a 6D wrench space. All forces acting on the object (including those from the friction cone) will result in a torque on the object. The torques must be in equilibrium as well as the forces, in order for the object to be in rest. This can be described as in the following equation for each contact point, where $f_i$ depicts the force that can be applied in a contact point, $\mu$ the friction coefficient and $n_i$ the contact normal in contact point $i$.

\begin{equation}
||f_i-(f_i \cdot n_i)n_i|| \le -\mu (f_i \cdot n_i)
\end{equation}
As a result of the forces torques are created formalized by the following equation.
\begin{equation}
\tau_i = (c_i-r)\times f_i
\end{equation}
Given the force and the torque a wrench can be written as the combination of both.
\begin{equation}
w_i=\begin{pmatrix}
f_i\\
(c_i-r)\times f_i
\end{pmatrix}
\end{equation}




The calculation of the friction cone is divided into calculating the magnitude of the friction force based on the normal force at the contact point and calculating the direction of the different parts of the cone. Due to obvious reasons an approximation of a cone is adopted. This means in practice that one direction vector is calculated and then rotated around the contact normal force, in steps according to the resolution that is wanted.

Given a friction coefficient, $\mu$, and a normal force, $n_i$, to a contact point, $i$, the magnitude of forces in the cone can be calculated: 
\begin{equation}
f_{cone}=n_i\mu
\end{equation}

In figure \ref{fig:cone:2} a 2D case is illustrated to show the basic principle of the force part of the wrench space. For each of the contact points a friction cone is calculated. By translating the contact points with friction cone to the same reference point and span a convex hull a representation of the forces that the grasp can withstand is found, see figure \ref{fig:cone:3}.




